Is following series convergent? [duplicate]

Question

If $$ \sum x_n$$ is divergent where $x_n$'s are non negative. For $p \geq 1$, is it true that
$$ \sum \frac{x_n}{(x_1 + \cdots +x_n)^p} $$ is convergent?

Answer 1

We assume that $x_n>0$, for all $n$. Otherwise we delete all the zero terms and the sum does not change.

For $p=1$ it is not, in general, true. Take for example $x_n=1$, for all $n$, and then $$ \sum_{n=1}^\infty\frac{x_n}{x_1+\cdots+x_n}=\sum_{n=1}^\infty\frac{1}{n}=\infty. $$

Let $p>1$ and set $s_n=x_1+\cdots+x_n$. Then $$ \sum_{k=2}^n\frac{x_k}{(x_1+\cdots+x_k)^p}=\sum_{k=2}^n\frac{s_k-s_{k-1}}{s^p_k} \le\sum_{k=2}^n\int_{s_{k-1}}^{s_k}\frac{dx}{x^p}=\int_{s_1}^{s_n}\frac{dx}{x^p}<\int_{s_1}^\infty\frac{dx}{x^p}=\frac{s_1^{1-p}}{p-1}. $$ Hence $$ \sum_{n=1}^\infty\frac{x_n}{(x_1+\cdots+n_k)^p}\le \frac{x_1}{x_1^p}+\frac{s_1^{1-p}}{p-1}. $$